In this article we will talk about the perpendicularity of a line and a plane. First, the definition of a line perpendicular to a plane is given, a graphic illustration and example are given, and the designation of a line perpendicular to a plane is shown. After this, the sign of perpendicularity of a straight line and a plane is formulated. Next, conditions are obtained that make it possible to prove the perpendicularity of a straight line and a plane, when the straight line and the plane are given by certain equations in rectangular system coordinates in three-dimensional space. In conclusion, detailed solutions to typical examples and problems are shown.

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Perpendicular straight line and plane - basic information.

We recommend that you first repeat the definition of perpendicular lines, since the definition of a line perpendicular to a plane is given through the perpendicularity of the lines.

Definition.

They say that line is perpendicular to the plane, if it is perpendicular to any line lying in this plane.

We can also say that a plane is perpendicular to a line, or a line and a plane are perpendicular.

To indicate perpendicularity, use an icon like “”. That is, if straight line c is perpendicular to the plane, then we can briefly write .

An example of a line perpendicular to a plane is the line along which two adjacent walls of a room intersect. This line is perpendicular to the plane and to the plane of the ceiling. A rope in a gym can also be considered as a straight line segment perpendicular to the plane of the floor.

In conclusion of this paragraph of the article, we note that if a straight line is perpendicular to a plane, then the angle between the straight line and the plane is considered equal to ninety degrees.

Perpendicularity of a straight line and a plane - a sign and conditions of perpendicularity.

In practice, the question often arises: “Are the given straight line and plane perpendicular?” To answer this there is sufficient condition for perpendicularity of a line and a plane, that is, such a condition, the fulfillment of which guarantees the perpendicularity of the straight line and the plane. This sufficient condition is called the sign of perpendicularity of a line and a plane. Let us formulate it in the form of a theorem.

Theorem.

For a given line and plane to be perpendicular, it is sufficient that the line be perpendicular to two intersecting lines lying in this plane.

You can see the proof of the sign of perpendicularity of a line and a plane in a geometry textbook for grades 10-11.

When solving problems of establishing the perpendicularity of a line and a plane, the following theorem is also often used.

Theorem.

If one of two parallel lines is perpendicular to a plane, then the second line is also perpendicular to the plane.

At school, many problems are considered, for the solution of which the sign of perpendicularity of a line and a plane is used, as well as the last theorem. We will not dwell on them here. In this paragraph of the article we will focus on the application of the following necessary and sufficient condition perpendicularity of a straight line and a plane.

This condition can be rewritten in the following form.

Let is the direction vector of line a, and is the normal vector of the plane. For the straight line a and the plane to be perpendicular, it is necessary and sufficient that And : , where t is some real number.

The proof of this necessary and sufficient condition for the perpendicularity of a line and a plane is based on the definitions of the direction vector of a line and the normal vector of a plane.

Obviously, this condition is convenient to use to prove the perpendicularity of a line and a plane, when the coordinates of the directing vector of the line and the coordinates of the normal vector of the plane in a fixed three-dimensional space can be easily found. This is true for cases when the coordinates of the points through which the plane and the line pass are given, as well as for cases when the line is determined by some equations of a line in space, and the plane is given by an equation of a plane of some type.

Let's look at solutions to several examples.

Example.

Prove the perpendicularity of the line and planes.

Solution.

We know that the numbers in the denominators of the canonical equations of a line in space are the corresponding coordinates of the direction vector of this line. Thus, - direct vector .

The coefficients of the variables x, y and z in the general equation of a plane are the coordinates of the normal vector of this plane, that is, is the normal vector of the plane.

Let us check the fulfillment of the necessary and sufficient condition for the perpendicularity of a line and a plane.

Because , then the vectors and are related by the relation , that is, they are collinear. Therefore, straight perpendicular to the plane.

Example.

Are the lines perpendicular? and plane.

Solution.

Let us find the direction vector of a given straight line and the normal vector of the plane in order to check whether the necessary and sufficient condition for the perpendicularity of the line and the plane is met.

The directing vector is straight is

In this lesson we will look at the perpendicularity of lines in space, the perpendicularity of a line and a plane, and parallel lines that are perpendicular to a plane.
First, we give the definition of two perpendicular lines in space and their designation. Let us consider and prove the lemma about parallel lines perpendicular to the third line. Next, we will give the definition of a line perpendicular to a plane, and consider the property of such a line, while remembering relative position straight and plane. Next we prove the straight line and converse theorem about two parallel lines perpendicular to the plane.
At the end of the lesson, we will solve two problems on the perpendicularity of lines in a parallelepiped and a tetrahedron.

Topic: Perpendicularity of a line and a plane

Lesson: Perpendicular lines in space. Parallel lines perpendicular to a plane

In this lesson we will look at the perpendicularity of lines in space, the perpendicularity of a line and a plane, and parallel lines that are perpendicular to a plane.

Definition. Two lines are called perpendicular if the angle between them is 90°.

Designation. .

Consider the straight lines A And b. Lines can intersect, cross, or be parallel. In order to construct an angle between them, you need to select a point and draw through it A, and a line parallel to the line b. Straight and intersecting. The angle between them is the angle between the lines A And b. If the angle is 90°, then straight A And b perpendicular.

If one of two parallel lines is perpendicular to the third line, then the other line is perpendicular to this line.

Proof:

Let two parallel lines be given A And b, and straight With, and . It is necessary to prove that .

Let's take an arbitrary point M. Through the point M draw a line parallel to the line A and a line parallel to the line c(Fig. 2). Then the angle AMS equals 90°.

Straight b parallel to the line A by condition, the line is parallel to the line A by construction. This means straight and b parallel.

We have, straight and b parallel, straight With and parallel in construction. So, the angle between the lines b And With - is the angle between straight lines and, that is, the angle AMS, equal to 90°. So it's straight b And With are perpendicular, which is what needed to be proven.

Definition. A line is called perpendicular to a plane if it is perpendicular to any line lying in this plane.

Designation. .

1. Geometry. Grades 10-11: textbook for students educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 pp.: ill.

Tasks 5, 6, 7 p. 54

2. Give the definition of perpendicularity of lines in space.

3. Equal sides AB And CD quadrangle ABCD perpendicular to some plane. Determine the type of quadrilateral.

4. The side of the triangle is perpendicular to some line A. Prove that one of the midlines of the triangle is perpendicular to the line A.

Lesson 3.2.1

Perpendicularity of lines.

Perpendicular and oblique.

Theorem of three perpendiculars.

Definition: Two lines in space are called perpendicular (mutually perpendicular) if the angle between them is 90 degrees.

Designation. .

Consider the straight lines A And b.

Lines can intersect, cross, or be parallel. In order to construct an angle between them, you need to select a point and draw a straight line a` through it, parallel to the straight line A, and a line b` parallel to the line b.

Lines a` and b` intersect. The angle between them is the angle between the lines A And b. If the angle is 90°, then straight a and b perpendicular.

Lemma: If one of two lines is perpendicular to the third line, then the other line is perpendicular to this line.

Proof:

Let's take an arbitrary point M. Through the point M draw a line a` parallel to the line A and a line c` parallel to the line c. Then the angle AMS equals 90°.

Straight b parallel to the line A by condition, line a` is parallel to line A by construction. This means that the straight lines a` and b parallel.

We have, straight and b parallel, straight With and parallel in construction. So, the angle between the lines b And With - this is the angle between lines a` and b`, that is, the angle AMS, equal to 90°. So it's straight b And With are perpendicular, which is what needed to be proven.

Perpendicularity of a line and a plane.

Definition: A line is called perpendicular to a plane if it is perpendicular to any line lying in this plane.

Property: If a line is perpendicular to a plane, then it intersects this plane.

(If a┴ α, then a∩ α.)

Reminder. A straight line and a plane either intersect at one point, or are parallel, or the straight line lies in the plane.

Properties of perpendicular lines and planes:

Theorem: If one of two parallel lines is perpendicular to a plane, then the other line is also perpendicular to this plane.

In the first lesson, we studied the Lemma - if one of the parallel lines intersects the plane, then the other parallel line intersects the plane. Straight A intersects at an angle of 90 0, that is, perpendicular, then the other parallel line is perpendicular

Theorem: If two lines are perpendicular to a plane, then they are parallel.

Sign of perpendicularity of a line and a plane

Theorem: If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to the plane

Theorem: Through any point in space there passes a straight line perpendicular to a given plane and, moreover, only one.

Video tutorial 2: Theorem of three perpendiculars. Theory

Video tutorial 3: Theorem of three perpendiculars. Task

Lecture: Perpendicularity of a straight line and a plane, signs and properties; perpendicular and oblique; three perpendicular theorem

Perpendicularity of a line and a plane

Let's remember what perpendicularity of lines actually is. Lines that intersect at an angle of 90 degrees are perpendicular. In this case, the angle between them can be either in the case of intersection at some point or in the case of crossing. If some lines intersect at right angles, then they can also be called perpendicular lines if, thanks to parallel transfer the straight line is transferred to a point on the second straight line.

Definition: If a line is perpendicular to any line that belongs to a plane, then it can be considered perpendicular to this plane.

Sign: If on a certain plane there are two perpendicular lines and some third line is perpendicular to each of them, then this third line is perpendicular to the plane.

Properties:

• If some lines are perpendicular to one plane, then they are mutually parallel to each other.

• If there are two parallel planes, as well as some straight line that is perpendicular to one of the planes, then it is also perpendicular to the second.
• It is also possible to make the opposite statement: if a certain line is perpendicular to two different planes, then such planes are necessarily parallel.

Inclined

If some straight line connects an arbitrary point that does not lie on the plane with any point on the plane, then such a straight line will be called inclined.

Please note that it is inclined only if the angle between it and the plane is not 90 degrees.

In the figure, AB is inclined to the α plane. In this case, point B is called the base of the inclined one.

If we draw a segment from point A to the plane, which will make an angle of 90 degrees with the plane, then this segment will be called a perpendicular. Perpendicular is also called the shortest distance to a plane.

AC is a perpendicular drawn from point A to plane α. In this case, point C is called the base of the perpendicular.

If in this drawing we draw a segment that will connect the base of the perpendicular (C) with the base of the inclined one (B), then the resulting segment will be called projection.

As a result of simple constructions we got right triangle. In this triangle, angle ABC is called the angle between the oblique and the projection.

Three Perpendicular Theorem

Definition. A straight intersecting plane is called perpendicular to this plane if it is perpendicular to any straight line that lies in the given plane and passes through the point of intersection.
Sign perpendicularity of a straight line and a plane. If a line is perpendicular to two intersecting lines of a plane, then it is perpendicular to this plane.
Proof. Let A– straight line perpendicular to straight lines b And With belonging to the plane a. A is the point of intersection of the lines. In plane a draw a straight line through point A d, not coinciding with straight lines b And With. Now on plane a let's make a direct k, intersecting the lines d And With and not passing through point A. The intersection points are D, B and C, respectively. Let us plot it on a straight line A V different sides from point A there are equal segments AA 1 and AA 2. Triangle A 1 CA 2 is isosceles, because the height AC is also the median (feature 1), i.e. A 1 C=CA 2. Similarly, in triangle A 1 BA 2 sides A 1 B and BA 2 are equal. Therefore, triangles A 1 BC and A 2 BC are equal according to the third criterion. Therefore, angles A 1 BC and A 2 BC are equal. This means that triangles A 1 BD and A 2 BD are equal according to the first criterion. Therefore, A 1 D and A 2 D. Hence the triangle A 1 DA 2 is isosceles by definition. In an isosceles triangle A 1 D A 2 D A is the median (by construction), and therefore the height, that is, the angle A 1 AD is straight, and therefore a straight line A perpendicular to a straight line d. Thus it can be proven that the straight line A perpendicular to any line passing through point A and belonging to the plane a. From the definition it follows that the straight line A perpendicular to the plane a.

Construction a straight line perpendicular to a given plane from a point taken outside this plane.
Let a- plane, A – the point from which the perpendicular must be lowered. Let's draw a straight line in the plane A. Through point A and straight line A let's draw a plane b(a straight line and a point define a plane, and only one). In plane b from point A we drop to a straight line A perpendicular AB. From point B to the plane a Let us restore the perpendicular and designate the straight line on which this perpendicular lies beyond With. Through segment AB and straight line With let's draw a plane g(two intersecting lines define a plane, and only one). In plane g from point A we drop to a straight line With perpendicular to AC. Let us prove that the segment AC is perpendicular to the plane b. Proof. Straight A perpendicular to straight lines With and AB (by construction), which means it is perpendicular to the plane itself g, in which these two intersecting lines lie (based on the perpendicularity of the line and the plane). And since it is perpendicular to this plane, then it is perpendicular to any straight line in this plane, which means it is a straight line A perpendicular to AC. Line AC is perpendicular to two lines lying in the plane α: With(by construction) and A(according to what has been proven), it means that it is perpendicular to the plane α (based on the perpendicularity of the line and the plane)

Theorem 1 . If two intersecting lines are parallel to two perpendicular lines, then they are also perpendicular.
Proof. Let A And b- perpendicular lines, A 1 and b 1 - intersecting lines parallel to them. Let us prove that the straight lines A 1 and b 1 are perpendicular.
If straight A, b, A 1 and b 1 lie in the same plane, then they have the property specified in the theorem, as is known from planimetry.
Let us now assume that our lines do not lie in the same plane. Then straight A And b lie in some plane α, and the straight lines A 1 and b 1 - in some plane β. Based on the parallelism of planes, planes α and β are parallel. Let C be the point of intersection of the lines A And b, and C 1 - intersections of lines A 1 and b 1. Let us draw in the plane of parallel lines A And A A And A 1 at points A and A 1. In the plane of parallel lines b And b 1 line parallel to straight line CC 1. She will cross the lines b And b 1 at points B and B 1.
Quadrilaterals CAA 1 C 1 and SVV 1 C 1 are parallelograms, since their opposite sides are parallel. Quadrilateral ABC 1 A 1 is also a parallelogram. Its sides AA 1 and BB 1 are parallel, because each of them is parallel to the line CC 1. Thus, the quadrilateral lies in the plane passing through the parallel lines AA 1 and BB 1. And it intersects parallel planes α and β along parallel straight lines AB and A 1 B 1.
Since the opposite sides of a parallelogram are equal, then AB = A 1 B 1, AC = A 1 C 1, BC = B 1 C 1. According to the third sign of equality, triangles ABC and A 1 B 1 C 1 are equal. So, angle A 1 C 1 B 1, equal to angle ACB, is straight, i.e. straight A 1 and b 1 are perpendicular. Etc.

Properties perpendicular to a straight line and a plane.
Theorem 2 . If a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
Proof. Let A 1 and A 2 - two parallel lines and α - a plane perpendicular to the line A 1. Let us prove that this plane is perpendicular to the straight line A 2 .
Let's draw 2 intersections of a line through point A A 2 with plane α an arbitrary straight line With 2 in the α plane. Let us draw in the plane α through point A 1 the intersection of the line A 1 with plane α straight With 1 parallel to the line With 2. Since it's straight A 1 is perpendicular to the plane α, then straight lines A 1 and With 1 are perpendicular. And according to Theorem 1, the intersecting lines parallel to them A 2 and With 2 are also perpendicular. Thus, straight A 2 is perpendicular to any line With 2 in the α plane. And this means that straight A 2 is perpendicular to the plane α. The theorem has been proven.

Theorem 3 . Two lines perpendicular to the same plane are parallel to each other.
We have a plane α and two lines perpendicular to it A And b. Let's prove that A || b.
Through the points of intersection of the straight lines of the plane, draw a straight line With. Based on the characteristic we get A ^ c And b ^ c. Through straight lines A And b Let's draw a plane (two parallel lines define a plane, and only one). In this plane we have two parallel lines A And b and secant With. If the sum of internal one-sided angles is 180°, then the lines are parallel. We have just such a case - two right angles. That's why A || b.